Time Period of a damped physical oscillator

Question

If I have a stick that is oscillating in air, and due to damping will its period increase or decrease? Damping will reduce its angular velocity as it opposes the motion of the stick, however, won't damping also reduce the amplitude which causes it to travel less distance and thus take less amount of time?

Answer 1

If you have a general oscillator with restoring forces and damping forces:

$$F_{rest}=-kx,\text{ } F_{damp}=-b\frac{dx}{dt}$$

Such that the equation of motion would be:

$$m \frac{d^2x}{dt^2}=-b\frac{dx}{dt}-kx$$ $$m \frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0$$

This equation has solution of the form : $$Acos(\omega x + \theta)e^{t/\tau}$$

Where you can plug it back into the equation of motion and find: $$\tau=-\frac{2m}{b}$$ $$\omega=\sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$$ And the constants $A$ and $\theta$ depend on initial conditions. Now we found the frequency of oscillations for this damped oscillator; $\omega$. We can compare it to the undamped case; or harmonic case. The frequency of oscillations of a harmonic oscillator is $\sqrt{\frac{k}{m}}$. Now obviously $\frac{b^2}{4m^2}>0$, so $\omega_{damped}<\omega_{harmonic}$.

Due to damping; the frequency of oscillation decreases which is the same as saying its period increases.